Boundary layers of the Boltzmann equation in three-dimensional half-space. (English) Zbl 1487.35280
Summary: We consider the nonlinear boundary layers of the Boltzmann equation on a three-dimensional half-space by perturbing around a Maxwellian under the assumption that the Mach number of the Maxwellian satisfies \(\mathscr{M}_\infty < - 1\). In earlier works by S. Ukai et al. [Commun. Math. Phys. 244, No. 1, 99–109 (2004; Zbl 1083.82028)], nonlinear boundary layers of the Boltzmann equation in a half-line are considered, in a half-line are considered, the stationary solutions were obtained and nonlinear stability was confirmed. We establish the existence and uniqueness of stationary and time-periodic solutions for the three-dimensional half-space model and show that these solutions are asymptotically stable.
MSC:
| 35Q20 | Boltzmann equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
| 35B35 | Stability in context of PDEs |
| 35B10 | Periodic solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Keywords:
Boltzmann equation; boundary layers; time-periodic solution; stationary solution; asymptotic stabilityCitations:
Zbl 1083.82028References:
| [1] | Aoki, K.; Nishino, K.; Sone, Y.; Sugimoto, H., Numerical Analysis of Steady Flows of a Gas Condensing on or Evaporating from Its Plane Condensed Phase on the Basis of Kinetic Theory: Effect of Gas Motion Along the Condensed Phase, 35-85 (1993), Kinokuniya: Kinokuniya Tokyo · Zbl 0787.76089 |
| [2] | Bardos, C.; Caflisch, R. E.; Nicolaenko, B., The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas, Commun. Pure Appl. Math., 39, 323-352 (1986) · Zbl 0612.76088 |
| [3] | Bardos, C.; Golse, F.; Sone, Y., Half-space problems of the Boltzmann equation: a survey, J. Stat. Phys., 124, 275-300 (2006) · Zbl 1125.82013 |
| [4] | Carleman, T., Probleme Mathematiques dans la Theorie Cinetique des Gaz (1957), Almquist et Wiksell: Almquist et Wiksell Uppsala · Zbl 0077.23401 |
| [5] | Cercignani, C., Half-space problem in the kinetic theory of gases, (Kröner, E.; Kirchgässner, K., Trends in Applications of Pure Mathematics to Mechanics (1986), Springer-Verlag: Springer-Verlag Berlin), 35-50 · Zbl 0594.76068 |
| [6] | Chen, C.-C.; Liu, T.-P.; Yang, T., Existence of boundary layer solutions to the Boltzmann equation, Anal. Appl., 2, 337-363 (2004) · Zbl 1172.35358 |
| [7] | Coron, F.; Golse, F.; Sulem, C., A classification of well-posed kinetic layer problems, Commun. Pure Appl. Math., 41, 409-435 (1988) · Zbl 0632.76088 |
| [8] | Glassey, R. T., The Cauchy Problem in Kinetic Theory (1996), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA · Zbl 0858.76001 |
| [9] | Golse, F., Analysis of the boundary layer equation in the kinetic theory of gases, Bull. Inst. Math. Acad. Sin. (N.S.), 3, 211-242 (2008) · Zbl 1142.82360 |
| [10] | Golse, F.; Pertham, B.; Sulem, C., On a boundary layer problem for the nonlinear Boltzmann equation, Arch. Ration. Mech. Anal., 103, 81-96 (1988) · Zbl 0668.76089 |
| [11] | Golse, F.; Poupaud, F., Stationary solutions of the linearized Boltzmann equation in a half-space, Math. Methods Appl. Sci., 11, 483-502 (1989) · Zbl 0679.35021 |
| [12] | Grad, H., Asymptotic theory of the Boltzmann equation, (Laurmann, J. A., Rarefied Gas Dynamics, vol. 1, 26 (1963), Academic Press: Academic Press New York), 26-59 · Zbl 0115.45006 |
| [13] | Hilbert, D., Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen (1953), Chelsea Publishing Company: Chelsea Publishing Company New York, N.Y., xxvi+282 · Zbl 0050.10201 |
| [14] | Sone, Y., Kinetic theory of evaporation and condensation –linear and nonlinear problems-, J. Phys. Soc. Jpn., 45, 1 (1978) |
| [15] | Sone, Y., Kinetic Theory and Fluid Dynamics (2002), Birkhäuser: Birkhäuser Berlin · Zbl 1021.76002 |
| [16] | Sone, Y.; Aoki, K.; Yamashita, I., A study of unsteady strong condensation on a plane condensed phase with special interest in formation of steady profile, (Boffi, V.; Cercignani, C., Rarefied Gas Dynamics, vol. II (1986), Teubner: Teubner Stuttgart), 323-333 · Zbl 0607.76100 |
| [17] | Sun, J.; Tian, Q.-Z., The nonlinear boundary layer to the Boltzmann equation with mixed boundary conditions for hard potentials, J. Math. Anal. Appl., 375, 725-737 (2011) · Zbl 1206.35190 |
| [18] | Sun, J.; Tian, Q.-Z., The nonlinear boundary layer to the Boltzmann equation for cutoff soft potential with physical boundary condition, Nonlinear Anal., Real World Appl., 12, 3207-3223 (2011) · Zbl 1231.35140 |
| [19] | Suzuki, M.; Takayama, M., Stability and existence of stationary solutions to the Euler-Poisson equations in a domain with a curved boundary, Arch. Ration. Mech. Anal., 239, 357-387 (2021) · Zbl 1456.82888 |
| [20] | Suzuki, M.; Zhang, K. Z., Stationary flows for compressible viscous fluid in a perturbed half-space, Comm Math. Phys., 388, 1131-1180 (2021) · Zbl 1478.35168 |
| [21] | Tian, Q.-Z., Existence of nonlinear boundary layer solution to the Boltzmann equation with physical boundary conditions, J. Math. Anal. Appl., 356, 42-59 (2009) · Zbl 1195.35008 |
| [22] | Tian, Q.-Z.; Sun, J., Nonlinear stability of boundary layer solution to the Boltzmann equation with diffusive effect at the boundary, J. Math. Phys., 50, Article 103303 pp. (2009) · Zbl 1283.35060 |
| [23] | Ukai, S.; Yang, T.; Yu, S.-H., Nonlinear boundary layers of the Boltzmann equation: I. Existence, Commun. Math. Phys., 236, 373-393 (2003) · Zbl 1041.82017 |
| [24] | Ukai, S.; Yang, T.; Yu, S.-H., Nonlinear stability of boundary layers of the Boltzmann equation, I. The case \(\mathcal{M}_\infty < - 1\), Commun. Math. Phys., 244, 99-109 (2004) · Zbl 1083.82028 |
| [25] | Valli, A., Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), 10, 607-647 (1983) · Zbl 0542.35062 |
| [26] | Wang, W.-K.; Yang, T.; Yang, X.-F., Nonlinear stability of boundary layers of the Boltzmann equation for cutoff hard potentials, J. Math. Phys., 47, Article 083301 pp. (2006) · Zbl 1112.76039 |
| [27] | Wang, W.-K.; Yang, T.; Yang, X.-F., Existence of boundary layers to the Boltzmann equation with cutoff soft potentials, J. Math. Phys., 48, Article 073304 pp. (2007) · Zbl 1144.81422 |
| [28] | Yang, X.-F., Nonlinear stability of boundary layers for the Boltzmann equation with cutoff soft potentials, J. Math. Anal. Appl., 345, 941-953 (2008) · Zbl 1178.35062 |
| [29] | Yang, X.-F., The solutions for the boundary layer problem of Boltzmann equation in a half-space, J. Stat. Phys., 143, 168-196 (2011) · Zbl 1220.82095 |
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